Questions about sub-sigma-algebras and filtrations Let $X_i$ be a sequence of i.i.d. rv. When talking about $\sigma(X_1,\ldots)$, I understand this to mean the smallest sigma algebra under which $X_i$ are measurable. Formally, we can take $\mathbb{R}^\mathbb{N}$ and then look appropriate subsets of rectangles that generate $\sigma(X_1,\ldots)$. Now, suppose I'm trying to prove say, Kolmogorov 0-1 law. Now textbooks are talking about $\sigma(X_1,\ldots,X_k)$ and $B_k:=\sigma(X_{k+1},\ldots)$. Already I am confused. Surely what is tacit here is that these are some sort of cylinder projections from $\sigma(X_1,\ldots)$. After all, it wouldn't make any sense to even consider $\cap B_k$ right? 
Question 1: What is the formal definition of $B_k$. Is it a projection? In other words, do I just consider the smallest sigma algebra generated by $X_{k+1},\ldots$ and then slap on the necessary extra stuff on the left? 
 A: Some confusion may come from considering the 
product space $\mathbb{R}^\mathbb{Z}$,
rectangles, and projections as central to the problem. They are not.
Let's go back to the beginning. Your first sentence says "Let $X_i$ 
be a sequence of i.i.d. rv.". This means there is a probability 
space $(\Omega,{\cal F},\mathbb{P})$ and measurable maps
$X_i:(\Omega,{\cal F})\to (\mathbb{R},{\cal B}(\mathbb{R}))$ for $i=1,2,3\dots.$
These maps have other important properties, crucial to the proof of 
Kolmogorov's $0-1$ theorem, but let's ignore those for now.
You have correctly explained that $B_0:=\sigma(X_1,\ldots)$ 
is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{i\geq 1}$ measurable. 
Note that $B_0$ is a subset of $\cal F$.  Similarly, $\sigma(X_1,\ldots,X_k)$ 
is the smallest $\sigma$-algebra that makes all the maps $(X_i)_{1\leq i\leq k}$ measurable,
and $B_k$  is the smallest $\sigma$-algebra 
that makes all the maps $(X_i)_{i\geq k+1}$ measurable.
All of these are subsets of $\cal F$; in fact they are all subsets of $B_0$.
So, to address some of your questions:


*

*"After all, it wouldn't make any sense to even consider $\cap B_k$ right?"
Sure, why not? Each $B_k$ is a subset of $\cal F$, and  $\cap B_k$ means their intersection.

*"What is the formal definition of $B_k$?" Answer: $B_k$  is the smallest $\sigma$-algebra 
that makes all the maps $(X_i)_{i\geq k+1}$ measurable. 
This is the formal definition.

*"Is $B_k$ a projection?" No, $B_k$ is a subset of $\cal F$. 

*"In other words, do I just consider the smallest sigma algebra generated by $X_{k+1},\ldots$
and then slap on the necessary extra stuff on the left?"
Answer: $B_k$  is the smallest $\sigma$-algebra 
that makes all the maps $(X_i)_{i\geq k+1}$ measurable. Nothing needs to be added to it. 
It may be that I have completely missed what you are confused about. If so, let me know.
But this should get you started.     
